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8. APPLICATIONS: ESTIMATION OF TRANSPORT PROPERTIES 347
energy and size parameters in the potential energy relation
(i.e., Eq. 5.11). σ is in ˚ A, T and T c are in kelvin, and V c is in hydrodynamic theories that in liquid systems diffusion co-
efficient is inversely proportional to viscosity of solvent. For
3
cm /mol. The correlations for calculation of Lennard–Jones example, based on the hydrodynamic theory and the Stokes–
(LJ) parameters (ε and σ) from critical constants as given in Einstein equation, Wilke and Chang developed the following
Eq. (8.57) were developed by Stiel and Thodos [36]. There are relation for estimation of diffusion coefficient at infinite dilu-
some other correlations given in the literature for calculation tion [18, 28]:
of LJ parameters [18]. Typical values of ε and σ determined ( B M B) 1/2 T
from various properties are given in Table 6.16. In the above (8.60) D ∞L = 7.4 × 10 −8 0.6
AB
relation low-pressure diffusivity can be calculated through di- μ B V A
∞L
viding (ρD AB) by ρ (= 83.14T/P) in which T is in kelvin and where D AB is the diffusion coefficient (in cm /s) of solute A in
2
◦
◦
2
P is in bar. Calculated D AB would be in cm /s. solvent B, when concentration of A is small (dilute solution).
For practical calculations, a more accurate estimation The superscript ∞ indicates the system is dilute in solute and
method is required. Most of these correlations are based on for this reason concentration of solute is not included in this
the modified version of Chapman–Enskog theory [18]. The equation. M B is the molecular weight of solvent (g/mol), T
is absolute temperature in kelvin, and μ B is the viscosity of
empirical correlation of Chen–Othmer for estimation of D AB
of gases at low pressures is in the following form [28]: solvent B (in cp). Because the solution is dilute, μ B is almost
1/2 the same as viscosity of solution. V A is the molar volume of

−2
1.518 × 10 T 1.81 1 + 1 solute A at its normal boiling point in cm /mol and it may
3
(8.58) D ◦ = M A M B
AB 0.1405 0.4 0.4 2 be calculated from V c according to the relation given in Eq.

P (T cA T cB) V + V
cA cA (8.59). ψ B is called association parameter for solvent where for
where D AB is the diffusivity of A in B at low pressures in cm /s, water the value of 2.6 is recommended [28]. For methanol and
2
◦
T is in kelvin, P is in bar, M in g/mol, and T cA and V cA are the ethanol, ψ B is 1.9 and 1.5, respectively. For benzene, heptane,
3
critical temperature and volume of A in kelvin and cm /mol, and unassociated solvents (most hydrocarbons) its value is
respectively. This method can be used safely up to pressure of 1.0 [18, 28]. The average error for this equation for some 250
about 5 bar. This equation predicts self-diffusion coefficient systems is about 10% [18].
2
of methane at 298 K and 1 bar as 0.248 cm /s versus the value Another simple method derived from Tyn and Calus equa-
of 0.194 from the kinetic theory (Eq. 8.56). For hydrocarbon– tion and is given as follows [18]:
hydrocarbon systems the API-TDB recommends the Gilliland V 0.267 T
∞L
method in the following form [5]: (8.61) D AB = 8.93 × 10 −8 B 0.433
1/2 V A μ B

4.36 × 10 T 1.5 1 + 1 where the parameters and units are the same as those given in
−3
(8.59) D ◦ = M A M B
AB
1/3 1/3 2 Eq. (8.60). V B is the molar volume of solvent at its boiling point --`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
P V A + V B and can be calculated from V cB similar to V A . Equation (8.61)
V i = 0.285V 1.048 is suitable for organic and hydrocarbon systems. Because of
ci higher accuracy, the Wilke–Chang method (Eq. 8.60) is widely
where V i is the liquid molar volume of component i at its nor- used for calculation of diffusion coefficient of liquids and it
mal boiling point and V ci is the molar critical volume and both is also recommended in the API-TDB [5].
3
are in cm /mol. Other units are the same as in Eq. (8.58). This As shown in Table 8.8 diffusion coefficient of a binary liquid
equation can be used up to pressure of 35 bar with an accu- system depends on the concentration of solute. This is the rea-
racy of about 4% as reported in the API-TDB [5]. Several other son that most experimental data on liquid diffusivity are re-
methods for prediction of gas diffusivity at low pressures are ported for dilute solutions without concentration dependency
given by Poling et al. [18]. and for the same reason predictive methods (Eqs. (8.60) and
(8.61)) are developed for diffusion coefficients of dilute solu-
8.3.2 Diffusivity of Liquids at Low Pressures tions. There are a number of relations that are proposed to
L
calculate D AB at different concentrations. The Vignes method
∞L
Calculation of diffusion coefficients for liquids is less accurate suggests calculation of D AB from D ∞L and D BA as follows [35]:
AB
than gases as for any other physical property. This is mainly L ∞L x B ∞L x A

due to the lack of a perfect theory for liquids. Generally there (8.62) D AB = D AB D BA α AB
are three theories for diffusivity in liquids: (1) hydrodynamic where x A is the mole fraction of solute and x B is equal to 1 − x A .
theory, which usually applies to systems of solids dissolved in Parameter α AB is a dimensionless thermodynamic factor in-
liquids, (2) Eyring rate theory, and (3) the free-volume theory. dicating nonideality of a solution defined as
In the hydrodynamic theory, it is assumed that fluid slides
over a particle according to the Stoke’s law of motion. The (8.63) α AB = 1 + ∂ ln γ A = 1 + ∂ ln γ B
Eyring theory was presented earlier by Eq. (8.55) in which ∂ ln x A T,P ∂ ln x B T,P
molecules require an energy jump before being able to dif- where γ A is the activity coefficient of solute A and can be esti-
fuse. The free-volume theory says that for a molecule to jump mated from methods of Chapter 6. For ideal systems or dilute
to a higher energy level (activation energy), it needs a critical solutions (x A = 0), α AB = 1.0. For simplicity in calculations for
∼
free-volume (V ) and Eq. (8.55) can be modified by multiply- hydrocarbon–hydrocarbon systems this parameter is taken as
∗
A
ing the right-hand side by factor exp V /V , where V is the unity.

∗
A
apparent molar volume of liquid. None of these theories is Another simple relation is suggested by Caldwell and Babb
perfect; however, it can be shown by both the Eyring rate and and is also recommended in the API-TDB for hydrocarbon
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